Question

Let $f:X\rightarrow Y$ be an arbitrary function and A $\subseteq$ X.
i) Show that A $\subseteq$ $f^-^1(f(A))$
ii) Give an example to show that in general A = $f^-^1(f(A))$ .

iii) Show that, if is injective, then A = $f^-^1(f(A))$
iv) Show that, if X and Y are modules; is a homomorphism of modules and A is a submodule of X such that ker$\subseteq$, then we also have A = $f^-^1(f(A))$

Answer 1

Asto fixy 0 Let is a function and at A be any element fast f A) ya = f(a) Set, o ya c Y # N A THAKAJ) f(f(A)) = fxEX: fale f (A)] Since, Yu = f cas e f (A) 7 at ff (f Al). Hence, Aç ft (f AJ) x= { 1, 2, 33, = {9163 A = $ 24, f: x 4 as f = a, f(3) = a, f(2)=b f(A) = (fon, fas} = {a, 63 - S! (fm) = {1, 2, 3) : A & ft (f A)). x= { 1, 2, 3, 4 = {a,b,c,d} f: x - 24 , A={1,33 f)= a, f2) =b, f²) = c. f AJ = {fo, fo} = (a. 13 gl ( f (A)) = (1, 33 = 4

LV ID A Assume that f is injective function. Let & RES ( f (A1) + fx & f (A) fel= fcas for some at A x=a e f is injective function XEA [er att and x-a] ft (f (A)) CA Hence, by past 6 A-st (fa). (1) Kerf & À A is submodule of & Let ac ft (fa) Baf ra f(x) E FCA) x = f(a) for some a CA f (x- a) = f(x) - feed = o. x-at ker(f) a-a E A en ker(f) & A x- a - b For some SEA → X - a+b Note that a bt A. Code atbcA es. A is submodule of X. & A. Hence, f (f (AJ) CA Thus by part 1, f(f(A)) = A